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Procedural Techniques for Simulating the Growth of Plant Leavesand Adapting Venation Patterns

Dr. Monssef Alsweis∗ Prof. Oliver Deussen†

Department of Informatics and Information ScienceUniversity of Konstanz

Figure 1: Our system allows for the simulation of various types of leaves including different venation patterns and complex leaf shapes

Abstract

This paper presents biologically-motivated a procedural method forthe simulation of leaf contour growth and venation development.We use a mathematical model for simulating the growth of a plantleaf. Leaf tissue is regarded as a viscous, incompressible fluidwhose 2D expansion is determined by a spatially varying growthrate. Visually realistic development is described by a growth func-tion RERG that reacts to hormone (auxin) sources embedded inthe leaf blade. The shape of the leaf is determined by a set of fea-ture points within the leaf contour. The contour is extracted fromphotos by utilizing a Curvature Scale Space (CSS) Corner Detec-tion Algorithm. Auxin transport is described by an initial auxin fluxfrom an auxin source to an auxin sink that is gradually channelizedinto cells with high levels of highly polarized transporters. The leafis presented as a triangulated double layer structure that consists ofa Voronoi-Diagram that is discretised along the vein structures.

CR Categories: I.3.3 [Computer Graphics]: Three-DimensionalGraphics and Realism—Display Algorithms I.3.7 [ComputerGraphics]: Three-Dimensional Graphics and Realism—Radiosity;

Keywords: Leaf development, auxin rate, curvature scalespace(CSS), animation, botanical simulation

1 Introduction

Realistic modeling of growing plant leaves has received only a lim-ited attention in computer graphics. This is interesting since it also

∗e-mail:monssef.alsweis@uni-konstanz.de†e-mail:oliver.deussen@uni-konstanz.de

has a long history in bridging biology, theoretical studies of mor-phogenesis, and visualization[Prusinkiewicz 1994]. In this paper amathematical model is presented to simulate the growth of leaves.The tissue of the leaf is regarded as a viscous, incompressible fluidwhose 2D expansion is caused by a non-zero growth rate that dif-fers locally [Wang et al. 2004]. We propose a physically-based ap-proach to model growth based on the corresponding expansion rate[Coley et al. 2006]. We ignore body forces such as gravity and onlyassume surface forces.

Leaf form and vascular patterns provide some of the most impres-sive examples of the complexity of biological shapes generated innature. In multicellular organisms, boundaries have the role of pre-venting the intermingling of two different cell populations and inorganizing the morphogenesis of organs and the entire organism.Plant leaves have two different cell populations, the adaxial (or up-per) and abaxial (or lower) cell populations, and the boundary isconsidered to be important for lamina growth [Nakata and Okada2013].

In this paper we will character the growth by growth tensor fieldat X and Y and Z coordinates [Hejnowicz and Romberger 1984;Nebelsick et al. 2001; E et al. 2004]. The growth tensor allows fullcharacterization of the rate of growth in length, area, and volume,as well as rates of angular change between elements, and of vortic-ity in the growing organ. The growth tensor is a generalization ofthe relative elementary rate of X Coordinate growth RERGx, YCoordinate growthRERGy , and Z Coordinate growthRERGz .

Furthermore, the morphology of leaves is highly determined bywide variety in leaf venation patterns that has been classified forexample in [Nebelsick et al. 2001]. We use this system and adoptits terminology. This classification system does not only considerthe geometric arrangement of different vein classes but also theirrelation to other architectural features of leaves.

Figure 2 gives an overview of our proposed method. We simulatethe interplay between different processes and use a mathematicalmodel to simulate the growth of a plant leaf. (i) : we developed amethod for classifying a plant leaf by utilizing features of the leafcontour. A simple image of the contour can be extracted by utilizinga Curvature Scale Space (CSS) Corner Detection Algorithm. (ii) :

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A model generates visually realistic development using the growthfunction RERG that lets the leaf grow towards sources of the hor-mone Auxin which are embedded in leaf blade. The growth rate ofplant organ a scalar of a quantity vector, that depends on the levelof rigor with which we scrutinize our perception of growth. (iii): a auxin transport canalization-based model describes a processin which an initial auxin flows from a source to a sink while it isgradually canalized into files of cells with high levels of highly po-larized transporters. Our model is expressed in geometric terms anduses proximity criteria to determine new vein locations. (iv) : Theleaf is presented as a triangulated double layer structure that con-sists of a Voronoi-Diagram discretized along the vein structure andits corresponding diagram. The leaf mesh is mapped to deformedsaddle-like mid-surface and rippled contour.

2 Related Work

Related work for our approach encompasses biology as well asmodeling methods in computer graphics.

Biological Modeling: In literature, repeatedly a basic module wasused to simulate the development of common feature of the leaflamina and vein networks. The vascular system of plants consistsof a network of cell files (vascular strands) that extends through allorgans [Scarpella et al. 2010; AG and P 2005; Bilsborough et al.2011]. Over the past 20 years, genetic approaches have led to sub-stantial increase in our understanding of leaf and vascular devel-opment, and have provided good evidence that the growth regula-tor auxin provides important spatial cues for this. Since inhibitionof auxin transport affects the formation venation patterns, auxin islikely to be part of the involved signal [Scarpella et al. 2010]. Inthis paper we want to utilize auxin for the visual formation of plantleaves.

Haiyi et al. [Liang and L 2009] produce a model that uses geomet-ric and growth control parameters to determine the shape of finitelaminae. This allows for a comparative study of elongated leaf mor-phology. In [Liang and L 2009] a shape space for a growing elasticleaf is designed by using a combination of scaling concepts, stabil-ity analysis, and numerical simulations. This combination is con-sidered as increased relative growth strain. A long flat lamina de-forms to a saddle shape and/or develops undulations that may leadto strongly localized ripples as the growth strain is localized to thecontour of the leaf. Haiyi et al. [Liang and Mahadevan 2011] usea combination of surgical manipulations and quantitative measure-ments to confirm this hypothesis and provide a simple theory forchanges in the shape of a doubly curved thin elastic shell subject todifferential growth across its plan-form. This functional morphol-ogy suggests new bio-mimetic designs for deployable structures us-ing boundary or edge actuation rather than the usual bulk or surfaceactuation.

Hang et al. [Xiao and Chen 2011] establish phenomenologicalbuckling models to explain the curled configuration of driedleaves, where the driving force is the differential contraction strainfield. In the minimalist model, through a systematic study, theaveraged buckling curvature is correlated with the aspect ratioand normalized size of the leaf, as well as the magnitude ofthe differential strain. Hong et al. [ming Xu and jian He 2012]propose a method for modeling curly plant leaves that is basedon venation skeletons driving leaf surface deformation. A 2-Dleaf silhouette was extracted from a scanned leaf image. Thealgorithm computed the primary veins (medial axes) of a leaf,along which secondary veins were branched out automatically.SoHyeon et al. [Jeong et al. 2013] simulates the whole leaf surfaceto capture the fine details of desiccated leaves. In this paperwe will present the whole leaf surface as a triangulated double

layer structure that consists of a Voronoi-Diagram discreditedalong the vein structure corresponding diagram. The leaf meshis mapped to deformed saddle-like mid-surface and rippled contour.

In Computer Graphics, different approaches have been proposedfor modeling plants leaves: image-based modeling, particlesystems, implicit contours, and L-systems [Peyrat et al. 2008].

Image-based modeling: a first paper for modeling leaves usesimage-based modeling. Such methods reproduce a real shape ofa leaf provided by the user. Quan et al. [Quan et al. 2006] use asemi-automatic technique for modeling plants leaves directly fromimages. The approach has the advantage that the resulting modelinherits the realistic shape and complexity of a real plant. The userprovides several pictures taken from different angles, then a pointcloud is built and the segmentation of individual leaves (via a graph)is done with the image information. After this the user can manuallyrefine the segmentation in order to bypass errors due to overlappingleaves. A generic and deformable leaf model is then applied basedon this information.

Particle systems: Rodkaew et al. [Rodkaew et al. 2004] proposea particle transportation algorithm for modeling plants in differentcolors and with complex venation structures. The algorithm is initi-ated by randomly scattered particles inside the blade of a leaf. Eachparticle contains energy. A transportation rule directs each parti-cle toward a target. When particles are in close proximity, they arecombined. The trails of moving particles are used to generate thevenation patterns. Runions et al. introduced a biological algorithmin order to build leaves [Runions et al. 2005]; they use the shapeof the leaf, then build veins via simulation of hormone distributioninside the leaf.

Implicit contours: a method designed by Hammel et al. in [Ham-mel et al. 1992] models compound leaves using implicit contours.This model creates a planar scalar field in the proximity of the skele-ton. The margin is represented by a contour, defined as the locusof points with a given field value. The area bounded by this con-tour forms the surface of the leaf. Mundermann et al. in [Munder-mann et al. 2003] design a method for modeling lobed leaves. Thismethod extends the concept of sweeps to branched skeletons.

L-systems: The first methods to simulate certain patterns in na-ture found in plant development are L-systems [Prusinkiewicz andLindenmayer 1996; Rodkaew et al. 2002]. These methods arevery efficient to simulate plants organs and branching structures.Prusinkiewicz and Lindenmayer in [Prusinkiewicz and Linden-mayer 1996] proposed many methods based on L-systems. In [Ter-raz et al. 2009] an extension of L-systems is proposed, based onthree-dimensional (3D) generalized maps that allow an easier con-trol of the internal structure of 3D objects.

3 Simulation of Leaf Growth

Photosynthesis is a process used by plants and other organisms toconvert light energy coming from the sun into chemical energy thatcan be later released to fuel the organisms’ activities. In order toobtain realistic leaf growth, we develop a system using five pro-cesses to simulate leaf and leaf growth. The system is summarizedin Figure 2.

3.1 Leaf Type Modeling

The shape of the leaf blade and the type of leaf margin areimportant characteristics that help to identify plants (see Figure 3).Leaf blades vary to a large extent, they may be simple (apple, oak)

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Figure 2: Our system for leaf development.

or compound (divided into several smaller leaf like segments, asin honeylocust). The smaller segments are called leaflets and areattached to a stalk (rachis) with a petiolule. Leaflets can also bearranged palmately (horse chestnut) or pinnately (ash). Pinnatelycompound leaves are said to be odd pinnate (ash) when ending inone leaflet and even pinnate when ending in two leaflets (locust).This terminology is important in identifying plants by their leaves.

Figure 3: Different leaf shapes found in nature.

Figure 2 shows the schematic modeling processes for our leaves.Our modeling system provide the user with the ability to select oneleaf type from various given leaf classes. After that, the model-ing processes start to get the 2D silhouette of a leaf from a giveninput photography which is derived from the selected leaf type.Then the relative elementary axis-aligned growth rates RERGx

and RERGy are computed using the 2D silhouette. The relativeelementary growth rateRERGl(ζ) is the rate at which an infinites-imal distance ∆ζ, measured in the direction of growth line at a pointon growing silhouette, increases over time. RERG is mathemati-cally defined as [Hejnowicz and Romberger 1984]:

RERGl(ζ) = (1

∆ζ)(d∆ζ

dt) (1)

Using the relative elementary growth rate we create a growth curve(vector) for the X and Y coordinates of each leaf type (see Fig-ure4). The growth rate of a leaf on X or Y is a scalar or a vec-tor quantity. In order to produce different growth curves (vector)for same leaf type we used merge algorithm between two growthcurve(vector) which belong to same leaf type. For example, when ihave two growth rate vectorRERGx1 andRERGx2 for one leaftype, can i get new growth rate vectorRERGxn by apply a merge(inheritance) algorithm on both vectorRERGx1 andRERGx2.

Figure 4: Axis-aligned growth curves (vector) that define a leaf.

3.2 Leaf Expansion

To simulate leaf growth we use a two-dimensional model since thethickness of a leaf can be considered negligible compared to thegrowth of its surface. Under these assumptions, the surface Ex-pansion can be modeled by expansion rate as percent per day GR[Coley et al. 2006].

GR = 100 ∗ [e(ln(Sarea/Earea)/t) − 1] (2)

where Sarea and Earea are leaf area at two different measure-ments and t equals the number of days between measurements. Thegrowth model used in this paper has been tested on the growth ofseveral types of leaves. We observed that various parts of the leafslamina expand at different rates, depending on their distance fromthe tip and the age of the leaf. Figure 5(a) illustrates the growth of anatural leaf while in 5(b)-(e) computer simulations are shown usingthe expansion rate GR. Our results faithfully resemble the naturalgrowth.

3.3 Determination of Auxin Sources and Maxima

Our algorithm distributes Auxin sources at leaf edges and then wesimulate the venation process based on the Auxin level and con-sumption. Strong auxin sources are assumed to exist at edge loca-tions that grow stronger than others [Stanko et al. 2014]. Figure 6illustrates the relationship between the venation patterns and maxi-mum Auxin sources.

In this paper, we used the Shi-Tomasi corner detector and good fea-tures [Shi and Tomasi 1994] to track the max auxin placement atthe leaf contour which we get from the input photography. Thegrowth in each leaf corner must be larger than for other leaf loca-

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(a)(b)

(c) (d) (e)

Figure 5: Growth of a plant leaf in a time period. (a) Naturalgrowth of a plant leaf. (b-e) Computer simulated growth of a plantleaf.

tions. Therefore we deduce Auxin maxima to be located at leafcorners. In Figure 7 the red and blue circle represent such sources.

Figure 6: Illustration of the relationship between auxin maximaand primary and secondary venation processes.

3.4 Generating and Growing the Venation System

The plant growth regulator Auxin seems also to be responsible forthe development of the venation patterns [Scarpella et al. 2010].For most leaves, the primary and secondary veins are not only themost obvious features but also play the most significant role in leafdeformation. Therefore in this paper we concentrate on simulatinggrowth and development of primary and secondary veins. Our al-gorithm, however, can also simulate third and subsequent levels ofveins.

During initial leaf growth, a small primordium becomes visibleat the flanks of the SAM (the shoot apical meristem). EpidermalAuxin flow converges to form a maximum of Auxin activity at thetip of the primordium. It is drained through the center of the pri-mordium, marking the position of mid vein of the new leaf. In

(a)

(b)

Figure 7: Maxima for the Auxin distribution in several leaf types.

(a) (b) (c)

Figure 8: Development of primary veins with respect to localiza-tion of max auxin. In first row illustrated canalization process forprimary veins at first step. While the second row illustrated thedevelopment of primary vein after growth time.

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(a) (b)

Figure 9: Illustration of natural venation system and our venationsystem. (a) Natural leaf. (b) Computer simulated leaf venation.

primary morphogenesis, leaves grow predominantly via cell divi-sion to acquire their shape and vascular pattern. Auxin maxima atthe margins of the leaf correlate with sites of lateral vein formationand positions of serration development.

We generate the primary veins related to the given leaf type andthe maximum of auxin sources placement. Maximum of auxin areassumed at locations that are growth rate farther than a threshold.Then canalized between maximum auxin and the nodes that are re-lated on the leaf type. Figure 8 illustrates the development of pri-mary veins with respect to the localization of maximum of Auxinat leaf margin.

In order to generate the secondary veins, we firstly develop a func-tion to choose a parametric number n of nodes in each primaryvein. Then we find n Auxin sources in the leaf margin. Each Auxinsource is now assumed to join the vein node that is closest to it. InFigure 10 we simulate the development process of primary and sec-ondary veins with respect to localization of Auxin maxima at leafmargin.

For our needs, we developed an interface for automatic generatingvenation systems from a leaf skeleton. The Venation skeleton is in-troduced for 2D shapes in order to provide a symmetry-based shaperepresentation for perception and recognition, see Figure 10. Asshown in Figure 9 the natural venation systems in column (a) andour computer simulated venation systems in column (b) are quitesimilar.

(a)

(b)

(c)

(d)

Figure 10: Development of primary and secondary veins with re-spect to localization of max auxin for several leaf type.

Figure 11: Characterization of two layer leaf mesh.

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(a) (b)

Figure 12: Characterization of 2D and 3D deformation shape func-tion.

3.5 Building the Leaf Mesh

We have constructed a leaf skeleton with two boundary curves andmidvein curve. To mesh the void area within these boundary curves,we employ Delaunay triangulation scheme, because it can deal withthe problem of concave area in the leaf blade, see Figure 11. Forexample, lobed leaves often have irregular silhouette characterizedby a number of concave outline.In this paper each initial leaf is formed from two layers of meshesin order to regard the thickness of leaf. Generally, each vein-pointis inserted as a new vertex in the leaf mesh in order to link meshtriangles with veins system. The leaf mesh is textured with differ-ent colors such that each color is related to the position and the typeof the vertex. In addition to that, the leaf contour is prepared sep-arately, and then it is linked to leaf mesh. The objective is to getmore control on the deformation of the leaves contour and on theircolors.

3.6 3D Deformation Function

Leaves and flowers usually have a typical, saddle-like mid-surfaceand a characteristic rippling pattern at their edges. To produce aperfect saddle shape we propose the function

τ(x, y) = λ(x+ y)2 − µ(x− y)2 , (3)

where 0 <= λ <= 1 and 0 <= µ <= 1 are constants. It fulfillsall the arithmetic conditions [Liang and L 2009], and its plot has theshape of a saddle, as in Figure 12. To investigate the mechanisticbasis of rippling behavior and its physiological role during leavesgrowth, we use a function ϕ for periodic rippling [Liang and L2009]:

ϕ(x, y) = δ(y)sin(kx) . (4)

Here k is the dimensionless wave number, and δ(y) is the cross-sectional profile of the surface.

In Figure 12 we show the resulting 2D and 3D deformation shapesfrom saddle and rippling functions. In order to get the 3D deformedleaf surface we utilize the deformation function illustrated in Fig-ure 12. Our system can create several deformation shapes fromdifferent deformation function. In addition to that, each leaf typehave its own deformation function.

4 Modeling Results

In order to implement our method we programmed an intuitive sys-tem using Microsoft Visual Studio 2013 c++, OpenCV, OpenGL,and QT library.

Firstly, we produced the growth curve for several leave types. Fig-ure 4 shows different photos of growth curves used to simulate thegrowing of leaves. Eq. (1) represents a growth curve (DNA growth)on the x and y coordinates for each leaf type. the surface expansion

is modeled by expansion rate as percent per day GR, see Eq. (2).The growth of a plant leaf in a time period is simulated in Figure 5,which shows in (a) the natural growth of a plant leaf, and in (b-e)computer simulated growth of a plant leaf.

Our algorithm detects the placement of maximum auxin sources atleaf edges using Shi-Tomasi corner detector. Figure 7 illustrates thealgorithm to determine the max auxin distribution for several typesof leaves. After identifying the leaf type, max auxin placement,and number of primary veins we generate the primary venation de-velopment system for several leaf types. Figure 8 illustrates thedevelopment of primary veins with respect to localization of maxauxin at leaf margin. Our system uses a sophisticated function inorder to generate the secondary veins. The development of primaryand secondary veins is simulated in Figures 9 and 10.

We mesh the void area within boundary curves using Delaunay tri-angulation scheme to generate 3D leaf form. In order to regard thethickness of leaf we simulate each leaf with two layers of mesh, seeFigure 11. Our system creates several deformation shapes gener-ated from many different deformation functions used to deform theleaf shapes, see Figure 12. Then we texture the leaf with differentcolors, see Figure 1.

Compared to previous related work on modeling and simulatinggrowth of leaves, our method is motivated by all growth stage frombiology and ecology science in order to get a natural modeling ofseveral leaf development types.

5 Future Work

We presented a technique for constructing realistic leaf models forseveral leaf types; automatically, without any manual interactions.We showed how to model 2D natural growth for several types ofleaves, and we illustrated 3D renderings of deformed results of eachleaf Type. In the future we plan to develop our modeling systemto include the simulation of different outer or inner effects on theleaves. Various effects, such as cracks, insect attacks, and growingspots, will be incorporated into the simulation to improve realism.Finally there are many natural phenomena that happen with leaveswhich we plan to simulate.

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